Finite Group Databasebeta

← #46846513

Subgroups — #46846513

Order 34688.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774444C211yesmaximal, characteristic, complemented
2C4×#16864655C211yesmaximal, characteristic, complemented
3#31774224C211yesmaximal, characteristic, complemented
4#31774220C211yesmaximal, characteristic, complemented
5#31774180C211yesmaximal, characteristic, complemented
6#31774176C211yesmaximal, characteristic, complemented
7#31774095C211yesmaximal, characteristic, complemented
8#31774091C211yesmaximal, characteristic, complemented
9#31774045C211yesmaximal, characteristic, complemented
10#31774041C211yesmaximal, characteristic, complemented
11#31774010C211yesmaximal, characteristic, complemented
12#31774007C211yesmaximal, characteristic, complemented
13#31773995C211yesmaximal, characteristic, complemented
14#31773952C211yesmaximal, characteristic, complemented
15#440×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654—21no—
18C2×#16864641C2211yescharacteristic
19C2×D2168C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21#22576159C2211yescharacteristic
22#22576156C2211yescharacteristic
23#22576146—21no—
24#22576135C2211yescharacteristic
25#22576133—21no—
26#22576131C2211yescharacteristic
27Q8×Dic271C2211yescharacteristic
28C4×D1084—21no—
29C4×D1084C2211yescharacteristic
30C4×D1084C2211yescharacteristic
31#22576122—21no—
32#22576122C2211yescharacteristic
33C4×C22⋊D271—21no—
34#22576118—21no—
35#22576118C2211yescharacteristic
36C42×D271—21no—
37C42×D271C2211yescharacteristic
38#22576093C2211yescharacteristic
39#22576092C2211yescharacteristic
40#22576091C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576071C2211yescharacteristic
52C2×SD16×C271C2211yescharacteristic
53C4×Q8×C271C2211yescharacteristic
54C4×D4×C271C2211yescharacteristic
55#107×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#97×C271C2211yescharacteristic
59#16864655—22no—
60#16864655—22no—
61#16864654—44no—
62C2×Dic542—21no—
63C2×D1084—21no—
64C2×D1084—21no—
65C2×D1084C2311yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74C2×C4×D271D411yescharacteristic
75C2×C4×D271D411yescharacteristic
76C2×C4×D271C2311yescharacteristic
77#16864641—22no—
78D2168—22no—
79#16864639—22no—
80C2×C271⋊C8C2311yescharacteristic
81C2×C271⋊C8C2311yescharacteristic
82C4⋊Dic271—21no—
83C4⋊Dic271C2311yescharacteristic
84C4⋊Dic271C2311yescharacteristic
85Dic271⋊C4—21no—
86C22⋊Dic271—21no—
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92C4×Dic271—21no—
93C4×Dic271—21no—
94C4×Dic271D411yescharacteristic
95C4×Dic271D411yescharacteristic
96C4×Dic271C2311yescharacteristic
97C2×Q8×C271C2311yescharacteristic
98C2×D4×C271C2311yescharacteristic
99C22×C4×C271—21no—
100SD16×C271—22no—
101C2×C8×C271C2311yescharacteristic
102C2×C8×C271C2311yescharacteristic
103C4⋊C4×C271—21no—
104C4⋊C4×C271C2311yescharacteristic
105C4⋊C4×C271C2311yescharacteristic
106C22⋊C4×C271—21no—
107C42×C271—21no—
108C42×C271C2311yescharacteristic
109C22×D271—21no—
110C22×D271—21no—
111C22×D271C2×D411yescharacteristic
112Dic542—42no—
113D1084—24no—
114D1084—21no—
115D1084—21no—
116D1084—21no—
117D1084—42no—
118D1084#4112yescomplemented
119C22⋊D271—42no—
120C22⋊D271—42no—
121C22⋊D271—42no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C4×D271—24no—
128C4×D271—21no—
129C4×D271—21no—
130C4×D271—22no—
131C4×D271—41no—
132C4×D271—42no—
133C4×D271—42no—
134C4×D271—42no—
135C271⋊C8—21no—
136C271⋊C8—21no—
137C23×C271—21no—
138Q8×C271—21no—
139Q8×C271#4112yescomplemented
140D4×C271—21no—
141D4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—42no—
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2411yescharacteristic
149C8×C271—21no—
150C8×C271—21no—
151C22×C271—41no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271C22×D411yescharacteristic
155C2×D271—21no—
156C2×D271—22no—
157C2×D271—21no—
158C2×D271—21no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271—41no—
163C2×D271—41no—
164C4×C271—22no—
165C4×C271—21no—
166C4×C271—21no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271C2×#4111yescharacteristic
170C4×C271C2×#4111yescharacteristic
171Dic271—21no—
172Dic271—21no—
173Dic271—41no—
174Dic271—41no—
175Dic271AGL(1,ℤ/8)12yes—
176C2×C271—81no—
177C2×C271#52511yescharacteristic
178C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
179C2×C271#57511yescharacteristic
180D271—41no—
181D271—42no—
182D271—81no—
183C271#311511yescharacteristic, complemented, Sylow
184#3115—2711nomaximal, Sylow
185#532—2711no—
186C4×#41—2711no—
187#482—2711no—
188#482—2711no—
189#475—2711no—
190#474—2711no—
191#460—2711no—
192#459—2711no—
193#449—2711no—
194#449—2711no—
195#442—2711no—
196#442—2711no—
197#440—2711no—
198#440D27111yescharacteristic, complemented
199#432—2711no—
200C2×#41—2711no—
201C2×#41—5421no—
202C2×SD16—2711no—
203C2×SD16C2×D27111yescharacteristic
204C2×D8—2711no—
205C2×D8—2711no—
206C4⋊Q8—2711no—
207#127—2711no—
208#124—5421no—
209#121—2711no—
210#121—2711no—
211#121—5421no—
212C4×Q8—2711no—
213C4×Q8C2×D27111yescharacteristic
214C4×D4—2711no—
215C4×D4—2711no—
216C4×D4—2711no—
217C4×D4—5421no—
218C4×D4—5421no—
219C4×D4—5421no—
220C4×D4C2×D27111yescharacteristic
221#117—2711no—
222#117—5421no—
223C2×C42—2711no—
224C2×C42—5421no—
225#107—2711no—
226#107C2×D27111yescharacteristic
227#106—2711no—
228#106—2711no—
229C4⋊C8—2711no—
230C4⋊C8—2711no—
231#103—2711no—
232#103C2×D27111yescharacteristic
233#102—2711no—
234#102—2711no—
235#102—2711no—
236#102—2711no—
237#102—2711no—
238#102C2×D27111yescharacteristic
239C22⋊C8—2711no—
240C22⋊C8—2711no—
241#97—2711no—
242#97C2×D27111yescharacteristic
243#41—5422no—
244#41—5422no—
245#41—10844no—
246C2×Q8—5421no—
247C2×Q8C22×D27111yescharacteristic
248C2×D4—2711no—
249C2×D4—2711no—
250C2×D4—5421no—
251C2×D4—5421no—
252C2×D4—5421no—
253C2×D4—5421no—
254C2×D4—5421no—
255C2×D4C22×D27111yescharacteristic
256C22×C4—21no—
257C22×C4—2711no—
258C22×C4—2711no—
259C22×C4—2711no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4—5421no—
264SD16—22no—
265SD16—5422no—
266D8—5422no—
267D8—5422no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8C22×D27111yescharacteristic
271C2×C8C22×D27111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—5421no—
276C4⋊C4—5421no—
277C4⋊C4C22×D27111yescharacteristic
278C4⋊C4C22×D27111yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C42—21no—
287C42—2711no—
288C42—2711no—
289C42—2711no—
290C42—5421no—
291C42—5421no—
292C42C22×D27111yescharacteristic
293C23—21no—
294C23—2711no—
295C23—5421no—
296C23—5421no—
297Q8—21no—
298Q8—10842no—
299Q8#1686465412yescomplemented
300D4—21no—
301D4—21no—
302D4—2712no—
303D4—5424no—
304D4—5421no—
305D4—5421no—
306D4—5421no—
307D4—10842no—
308D4—10842no—
309D4—10842no—
310D4—10842no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—2711no—
316C2×C4—2711no—
317C2×C4—2711no—
318C2×C4—5422no—
319C2×C4—5424no—
320C2×C4—5421no—
321C2×C4—5421no—
322C2×C4—5421no—
323C2×C4—5421no—
324C2×C4—10841no—
325C2×C4—10842no—
326C2×C4—10842no—
327C2×C4—10842no—
328C2×C4D4×D27111yescharacteristic
329C2×C4D4×D27111yescharacteristic
330C2×C4C23×D27111yescharacteristic
331C8—21no—
332C8—21no—
333C8—5421no—
334C8—5421no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—5422no—
339C22—5421no—
340C22—5421no—
341C22—5421no—
342C22—10841no—
343C22—10841no—
344C22—10841no—
345C22—10841no—
346C22—10841no—
347C22C2×D4×D27111yescharacteristic
348C4—22no—
349C4—21no—
350C4—21no—
351C4—41no—
352C4—41no—
353C4—2712no—
354C4—5421no—
355C4—5421no—
356C4—10841no—
357C4—10841no—
358C4#41×D27111yescharacteristic
359C4C2×#1686465411yescharacteristic
360C2—81no—
361C2—10841no—
362C2—10842no—
363C2—21681no—
364C2#3177440611yescharacteristic
365C2AGL(1,ℤ/8)×D27111yescharacteristic
366C2#3177469611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#440×C271C2
socleC22×C271C22×D4
hypercenter#440D271
hypocenterC271#3115
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #31774444C2semidirect
2. C4×#16864655C2semidirect
3. #31774224C2semidirect
4. #31774220C2semidirect
5. #31774180C2semidirect
6. #31774176C2semidirect
7. #31774095C2semidirect
8. #31774091C2semidirect
9. #31774045C2semidirect
10. #31774041C2semidirect
11. #31774010C2semidirect
12. #31774007C2semidirect
13. #31773995C2semidirect
14. #31773952C2semidirect
15. #440×C271C2semidirect
118. D1084#41semidirect
139. Q8×C271#41semidirect
183. C271#3115semidirect
198. #440D271semidirect
299. Q8#16864654semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#317744442
C4×#168646552
#317742242
#317742202
#317741802
#317741762
#317740952
#317740912
#317740452
#317740412
#317740102
#317740072
#317739952
#317739522
#440×C2712
#3115271